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Full text of "Treatise On Analysis Vol-Ii"

2    PARTICULAR CASES AND EXAMPLES        251

and that there exists a positive nonzero G-invariant measure /u- on X. Let P (resp. C)
be a G -packing (resp. a G-covering) such that P and C are /n-integrable. Show that
ft(Q  /A(P). (Remark that ^t(C) ^  p(C n j  P) =  /^- r - C n P).)

seG                                   se G

(b)    Suppose that there exists a G-invariant distance d on X defining the topology
of X. Let A(G) denote the greatest lower bound of the numbers /u,(C), where C runs
through all integrable G-coverings of X. Let r be a real number >0 such that there
exists a point a e X for which ju,(B(#; r)) > A(G). Show that there exists s ^ e in G
such that d(a, s - a) < 2r.

(c)    Suppose that X is a locally compact group, /x a left Haar measure on X, and G a
denumerable subgroup acting on X by left translations. Show that if A is an integrable
subset of X such that fju(A) > A(G), then there exists s e G n AA"1 such that s ^ e.

(d)    With the same hypotheses as in (a), let F be a jie-integrable G -tessellation and let
Go be a subgroup of finite index h in G. If si9 . . . , sh are a system of representatives of
the right cosets of GO in G, show that F0    (J   St  F is a Go-tessellation.

lf<lr

(e)     Same hypotheses as in (a). Let/^ 0 be a /u,-integrable function on X. Show that
there exist two points a, b in X such that

f /(*) dp,(x)       and       ^(P)  f(s - b)  f /(

JX                                              see                 Jx

(*)

(Observe that if g ^ 0 is an integrable function and E an integrable subset of X, then
there exists c e E such that     g(x) dp,(x) ^ #(c)/x(E), and c' G E such that

2. PARTICULAR CASES AND EXAMPLES

(14.2.1)    On the additive group R, Lebesgue measure (13.1.4) is a Haar
measure (left and right, since R is commutative). This follows from the
formula for change of variable (8.7.4) applied to the function (p() =  + a,

which gives f + /(f + a) dt = f+ /(0 dt for all/e Jfc(R) and all a e R.

J  oo                                      J -~ oo

(14.2.2)    Now consider the multiplicative group R* of real numbers >0.
This is a locally compact commutative group ((13.18.4), (4.1.2), and (4.1.4))
For each function fe JTC(R*), there exists a compact interval [a, b] with
0 < a < b, containing the support of /. Hence for each interval [c, d] in

R* containing Supp(/), the integral  J (/(O dt)/t is defined and its value,

r 4* oo

which we denote by        (/(O dt)/t, is independent of the choice of interval

/* + OO

[c, d] containing Supp(/).  We  assert that /K|      (f(t) dt)ft is a  Haar

measure on R *. From the formula for change of variable (8.7.4), it follows
that, for each s > 0, D in R such that